Kirillov theory, Treves strata, Schrodinger equations and analytic hypoellipticity of sums of squares

We attach representations to non-symplectic stratum in the characteristic set of real vector fields. This leads to Schrodinger operators. The analysis of the solutions of these Schrodinger equations allows us to construct smooth, non-analytic solutions of the sub-elliptic operator formed by taking sums of squares of the real vector fields.